Physics Cup – TalTech 2021 – Pr. 4, Hint 4.

Continuing with the previous hint, since the initial standard deviation of the speed of the electrons is much bigger than 2u , once an electron has accumulated enough speed to overcome the barrier, its speed excess over the threshold is essentially a random value homogeneously distributed between 0 and 2u . At this point, the phase diagram (with p-axis being vertical) is a curve consisting of an horizontally elongated rectangle (corresponding to the motion in the shock wave – the newly added section), joined with a vertically elongated rectangle (corresponding to the motion outside the shock wave), all together forming a 90-degrees-rotated-T-shaped pattern. During the further motion of the shock, this shape will change (eventually evolving into just a rectangle), but its surface area, the adiabatic invariant, is conserved. If 2u is small, the newly added rectangular section has a negligible surface area, which might not be the case if 2u is bigger. Based on all this you can deduce the speed distribution of the electrons at the end of the process, and hence, the pressure exerted by them.

Please submit the solution of this problem via e-mail to

For full regulations, see the “Participate” tab.